Reprinted from JoURNAL oF C0MBNATouAL TrsoRy, Serics AAll Rights Rcerved by A@demic Pr6s, New York and LondoD

Vot.63, No. 1, May 1993Pfinted in Belgium

The Harmonic Logari thms andthe Binomial Formula

SmveN RoulN

Department of Mathematics, California State Uniuersily,Fuller ton, C alifornia 9 26 3 4

Communicated by Gian-Carlo Rota

Received March 6. 1991

1. INrnopucrroN

The algebra I of polynomials in a single variable x provides a simplesetting in which to do the "polynomial" calculus. Besides its being analgebra, one of the nicest features of I is that it is closed under bothdifferentiation and antidifferentiation. That is to say, the derivative of apolynomial is another polynomial, and the antiderivative of a polynomialis another polynomial (provided we ignore the arbitrary constant).

Furthermore, within the algebra 9, we have the well-known binomialformula

( x + a 7 ' : n e Z , n 2 0 .

This formula may have been known as early as about 1100 ,ro, in theworks of Omar Khayyam. (Euclid knew the formula fog n:2 around300 nc). To be sure, the formula, as we know it today, was stated by Pascalin his Traiti du Triangle Arithmdtique in 1665.

But now suppose we wish to include the negative powers of x in oursetting. One possibility is to combine the positive and negative powersof x, by working in the algebra ,il of Laurent series of the form

o!-*oo*oThe algebra ,4 is certainly closed under differentiation, and there is evena binomial formula for negatiue integral powers,

( x + a 7 " : n e Z , n < 0 ,

0097-3165/93 $5.00Copyright @ 1993 by Acadcmic Press, Inc.

All rights of reproduction in any fom resryed.

( 1 )-f,

(;) ooxn-k,

(2)i.(;)

akx'-k,

r43

t44 STEVEN ROMAN

due to Newton, which converges for |xl> lal. (The binomial formula alsoholds for noninteger values of z, but we restrict attention to integer valuesin this paper.)

The algebra il does suffer from one drawback, however. It is not closedunder antidifferentiation. For there is no Laurent series /(x) with theproperty that Df(x):x-1. To correct this problem, we must introduce thelogarithm log x. As we will see, doing so produces some rather interestingconsequences. For it leads us to introduce some previously unstudiedfunctions, which Loeb and Rota have called the harmonic logarithms' Wealso obtain a generalization of the binomial coefftcients, and the binomialformulas (1) and (2), which holds for allintegers n. This generalization iscalled the logarithmic binomial formula, and has the form

).f;'(x + a):

where n is any integer, I is any positive integer, and the functions lli'6)

are the harmonic logarithms. When l:0, formula (3) reduces to thetraditional binomial formula (1), and when t:l and n L, we get newformulas. The coeflicients

are generalizations of the binomial coeflicients ([), and are defined for allintegers n and k. Loeb and Rota refer to these as the Romqn coefficients'

The first thorough study of the harmonic logarithms was made by Loeband Rota [3 ]. Our goal here is to report on some of the more basic aspectsof this study.

Before beginning, let us set some notation. The symbol D is used for thederivative with respect to the independent variable. Also, if P is a logicalrelation that is either true or false, we use the notation (P), due to lverson,to equal 0 if P is false and 1 if P is true. For example,

l x l : x ' ( - 1 ; t " o l .

2. THr Henrraouc Locmlrnnrs

We begin by letting Z be the set of all finite linear combinations, withreal coefficients, of terms of the form xt(log x)/, where i is any integer, and

7 is any nonnegative integer. That is, Z is the real vector space with basis

{ x'(log x)i I i, j e Z, i > 0\ . Under ordinary multiplication,

x '( log x)/ 'x ' ( log x) ' : x '* "( log x) '* ' ,

(3 )i.L;l

Af;\-ola.xk'

L;]

HARMONIC TOGARITHMS

Z becomes an algebra over the real numbers. Furthermore, the formula

Dxt( logx) i : ix i - t ( logx)/+7x'- t( log"; ;- t (4)

shows that Z is closed under differentiation. and the formulas

145

D - l x ' ( l o g * ) t : ; l ; x t * r ( l o g * ) i - . i , ; D - r x i ( l o g x ) i - ' ,t f l i + l

D- | x . ' ( log x ) / : + ( log x ; r * r' j + r '

can be used to give an inductive proof showing that Z isantidifferentiation. In fact, we can characterize Z as follows.

i * - r

(5)

closed under

Pnoposnrox 2.1. The algebra L is the smallest algebra that containsboth x and x-1, and is closed under dffirentiation and antidffirentiation.

Formulas (4) and (5) indicate that, while the basis {x'(log x)r} may besuitable for studying the algebraic properties of L, it is not ideal forstudying the properties of Z that are related to the operators D and D-t.To search for a more suitable basis for I, let us take another look at howthe derivative acts on powers of x. If we let

. /nr, \ ( x" for n)0,r,;'tx): t0 for n < 0

then the derivative behaves as follows.

Dlto)@) = n)'f! r(x),

for all integers n. Thinking of the functions ,ilo)(r) as a doubly infinitesequence, as shown in Fig. 1, we see that applying the derivative operatorD has the ellect of shifting one position to the left, and multiplying by aconstant.

Let us introduce the notation

f o r n * 0for n :0 .

...{3)(.) r13)G) )(p(*) .r[0)1x; r[0)1x; rl0)1x; r[0)1x; ...

0 0 0 1 x x 2 x 3

Frcunr I

(nt _n t :

[ t

t46 STEVEN ROMAN

Then the functions ,t!0)(x) are uniquely defined by the following properties:

(1) , i [o)(x): I

(2) Alo\(r) has no constant term for n t 0

(3) Dlro)(x): [nl ,1!01,(x).

Note that the antiderivative behaves nicely on the functions 1!|(r), exceptwhen applieO to,igl(x). With the understanding that D-t produces no

arbitrary constant terms, we can write

( t

D-lAf,ot1x1: l;i L'fl'(ol for n* -r

[0 for n: -1.

Following these guidelines, we can introduce a second row of functions),!,t,(*) into Fig. 1, by starting with ,i$)("):logx, and using conditionssimilar to conditions (1F(3). In particular, the conditions

(4) ,1$)(") : log x

(5) Alt)(x) has no constant term

(6) il"ft(u") :Snl )"f;!,(x)

uniquely define a doubly infinite sequence of functions ).f\(r), as shown in

Fig.2.This figure makes it rather easy to guess the general form of the

functions 1!]\(*).

PnopostrloN 2.2. The functions ,lf;)(x), uniquely defined by conditions(4)-(6) aboue, are giuen by

a,- ,, ... .ri!)1*; .lJ!)1xy r(l)G) .r6o)G) r{o)G) rt0)1x; r[0)1x;

0 0 0 1 * * 2 x 3

n,. e, ... rjl)1*; r(;)(*) {l)t'l .rf )t.l r{1)1x; rf)1*; rf)t*)

x-3 x -2 x - l logx x ( logx-1) x2( logx-1- | ) '3 ( togx-1- | - | ) " '

Ftcuru 2

1y(,\: {;:,", ,_ h.)

fr,, :arr,w h e r e h n : l * t l 2 + l l 3 + " ' * l l n f o r n > 0 a n d h o : Q '

HARMoNTc LocARrrHMs 147

Proof. Conditions 4 and 5 are clearly satisfied. As for condition 6, forn 0, we have

Finally, for n:0, we have

D( log x ) = x - l : LOl x - t . I

Note that the behavior of D-r on the functions ),lt)(x) is even nicer thanit is on the functions .i!o)(x), for assuming no arbitrary constant, we have

ID- | ilttlxl:

Lri ,I rll ,(*).

The vector space formed by using the functions ,1.!011x; and ,lf )(x) as abasis is closed under differentiation and antidifferentiation, but it is not analgebra. (The functions (logx)', for t>1, are not in this vector space, forinstance.) This prompts us to enlarge our class of functions as follows.

DnnNrnou. For all integers n and nonnegative integers t, we define theharmonic logarithms )'f'(r\ of order t and degree r as the unique functionssatisfying the following properties:

(1 ) l t t@) : ( log x ) '

(2) )"li'Q) has no constant term, except that l.[or(x;: 1

(3) D). f ' (x): ln l ) , ! ' - , (x).

This definition allows us (at least in theory) to construct the harmoniclogarithms by starting each "row" (that is, the harmonic logarithms of afixed order) at 1{\(x): (log x)'. Then we dilferentiate to get ,l'f)(x) forn < 0, and antidifferentiate to get A!)(x) for n > 0. In fact, with the usualunderstanding about D-r, we can write

l i )@l:an.,D-n( logx) '

Dx"(log x - h,) : t4xn - 1(log x - h,) + x" - |

: n x n - ' ( b g r - l r , + 1 )\ n )

: n x n - 1 ( l o g x - h , _ r )

: ln lx , - t ( logx -h ,_ r ) .

(6)

148 srEvEN RoMAN

where the an,t ate constants. In order to determine these constants, we hrst

observe that according to Property (1) of the definition,

( log x) ' : , t f ) (x) : ao.,( log x) '

and so eo,,:1. Property (3) tells us that

Ln1a, - t . ,D-@- t )1 log x ) ' : ln l ) ' f \ - r (x ) : D19(x) : a , . ,D- ( ' -

1 ) ( log x ) '

and so

ar , , :Ln1 an- r , t .

Thus. for n ) 0, we have

a n , t : n e n - r , , : t l ( t l - l ) a n - r , , : i " : n ( n - l ) " ' ( 1 ) a s , t : n l ,

and for n

HARMoNTc LocARrrHMs 149

3. Tnr Nuunnns l_nl!

Some values of fnl! are given in the following table:

n - 6 - 5 - 4 - 3 * 2 - 1 0 1 2 3 4 5

l t l tLn l ! - t , ,o u

-E ,

- t I I 1 2 6 24 r2o

It is well known thatn!:f(n+ 1), for n)0, where f(z)is the Gammafunction. The numbers Lnl! can also be expressed in terms of the Gammafunction.

PnoposnoN 3.1. For all integers n,

( r@+r) . fo, n20L'l! : 1 R", f(zl for n

150 srEvEN RoMAN

Proof. For (a), if n>0, then [nl! :nl, and the result is well-known.F o r n : 0 , w e h a v e l 0 l [ 0 - 1 1 ! : 1 . 1 : 1 : L 0 ] ! . F i n a l l y , i f n < 0 , t h e nn- l

HARMoMC LocARrrHMs 151

The next proposition shows that these coefficients really do generalize thebinomial coeffrcients.

PnoposnloN 4.1. Wheneuer n2k20, or k20>n, the Roman codficients agree with the ordinary binomial cofficients, that is,

l r l /n \Lr l: \r/

Proof. When n > k>-0, we have [n l! : n!, lkl! - lst., and ln - kll:(n - k)1, in which case the result follows directly from the definition. Fork>O>n, we have

l n l - - L , T t - 1 , - - . , , -Lk I Lklfr : /cl

: , . ln1Ln- 11"'Ln-k+ rf

1 , . , / n \: f i . ( n ) (n - 1 ) " ' (n -k * 1 ) : ( ; / I

As the next proposition shows, several of the algebraic properties ofthe Roman coellicients are generalizations of properties of the ordinarybinomial coelficients.

PnoposntoN 4.2. (a) For all integers n, k, and r,

l ' - l : l n IL k I l n - k l

(b) For all integers n, k, and r,

L;IL:]:L:]L;_;I(c) (Pascal's formula). For any two distinct, nonzero integers n and

k, we haue

l r l _ l n - 1 1 , l n - 1 - |L t l : L / . l * L r - t l

Proof. Parts (a) and (b) are direct consequences of the definition. Asfor part (c), the conditions on n and k are equivalent to the statementsLnl:n, Lk1:k, and ln-k1:n-k. Hence, using Proposition 3.2, wehave

r52 STEVEN ROMAN

I n- l1 l:--:-::-:------- -i-L'; 'l . L;: il : [ n * 1 l !lkltln- 1 -kl! Lk* |l!ln- kl!

[ n - 1 l ! 1 1 \- + - |tkf ln-k1lkLk- 1 l ! Ln- r - k l ! \L f t l ' Ln-k1

[ n - l l ! LnlLk - 1l! Ln * L - kl! lklln - kl

: . =.1 ' l ' , =, : l :1 ILkl! Lr - k l ! L^ |

Now let us consider some results that do not have analogsordinary binomial coefficients.

Pnoposnlox 4.3. For all integers n and k, we haue

k + ( n > k )

for the

I n f l k1 ( -1) '(a ' L t lL , l : g , - t1

L -;l : ( - 1)n*o*('>o)+'-"' Lf - ll(Knuth' s Rotation I Reflection Law)

( - 1 )o*,-", Lo-jrl : ( - 1)'�*,"", L/rl

(b)

(c)

Proof. To prove part

L;IL;]:(a), we use part (a) of Proposition 3.3,

Ln l! Lkl!Lkl!Ln -klt lnl l lk-nl l

1 ( - l ) n - k + ( n > k )

l n -k l l l k -n l t Ln-k f

To prove part (b), we use part (b) of Proposition 3.3,

l - n f L - n f l ( - l ) ' + ( n > o ) L / s - 1 1 ! 1

L-r l : L-ntL1r-4: Ln-n GTFGb;t1r_�r1

/ 1 \ n ), F & + ( r > o ) + t t t o l L k - 1 - l !\ - / [n - 1 l ! l k -n l l

: ( - 1 ) ' + & + ( u > o ) + t u ' o l I

k - 1 - 1 .

L r - | l .

I

HARMoNTc LocARTTHMS 153

As for part (c), we replace -kby k-lin part (b), to get

l , - ' , 1 : ( - l ) z + / < - 1 1 1 a 1 6 1 a 1 k - r < o ) | - k , 1 .

L k - l | '

L n - t l '

Using the fact that (-1)- t+(k-10), and rearranging, we getthe desired result. I

5. Tnr Loclnrrnulc BrNolrt,lr Fonuurl

Now let us turn to the logarithmic binomial formula. The next proposi-tion can be proved by induction using formulas (4) and (5).

PnoposntoN 5.1. Each harmonic logarithm )rf)(x) is a finite linearcombinqtion of terms of the form xi(logx)i, where i is any integer and j ishny nonnegatiue integer.

In view of Proposition 5.1, for any positive real number a, we canexpand the function ),f(x+a) in a Taylor series that is valid for lxlk, andllo)o@):0 for nk>0, formula (7) is justthe classical binomial formula (1).

Next, consider the case ,: 1 and n < 0. Since

),1\(x): { ' . ( tot x - h ') for n} 'o '

' lx" for n

154 srEvEN RoMAN

Interchanging the roles of x and a, and noting that L?l: ([) whenk>O> n, we get the classical binomial formula (2). (Proposition 5.2 tells usonly that this is valid for x > a, ralher than lxl > a.) Thus, we see that thelogarithmic binomial formula is indeed a generalization of the classicalbinomial formulas (1) and (2).

When n > 0, we may extend the definition of the harmonic logarithms oforder 1 by taking

,tf )(0) :,llt* ,tf )1x;: g.

When t:1 and n)-0,the piecewise definition of ,lf;r(x) suggests that wesplit the sum on the right side of (7), to get

).? (x + o, : i,li1 ̂ f, rr'r,o * o I *,1i1" - r *

= i. (;) )t)o@) xk + a'-:i., L;l(r-valid for lxlal0. This form is convenient for determiningconvergence on the boundary.

Lnuue 5.3. Let a>0. Consider the series

S l ' l i l \ -o:X*,Lr l \" /

'

(1) For n>0, this series conuerges for all lxl (4.

(21 For n:0, this series conuergesfor all lxl (4, except x: -a.

Proof. For k > n20, we have

I r l L n l ! n t ( k - n - l ) t ' t 1 \ k - n - lI r : --- : --- : - : - t --------r

Lk | [ /< l ! [n - k l ! k ! ( - 11^- ' - '

Therefore, if n > 0, and l.xl ( a, we have

k ( k - r ) . . . ( k - n ) '

l l n l lx \ t I n l n l

lL* l l ; / |

HARMoNTc LocARrrr{Ms 155

and it is well known that this logarithmic series converges for lxlal{I,except at xf a: -1, or x: -e. (See, for example, 12, p.2l3l.) I

Abel's limit theorem 12, p.l77l now allows us to deduce the followingproposition.

PnopostrroN 5.4. The logarithmic binomial formula of order 1

)", t1x+o): i l !_]r. , ; ' . r1oy*r, (8)' EoLk I

with a>0, is ualid.

(1) For lx l 0, this is

(x * 1 )" [log(x + 1 ) - 0,, : L(ir), - o, - r, "* * * ;-, L;-l

"-

which is valid for lxl( 1, where the left hand side is 0 for x: -1.Rearranging terms, we get the following expansion.

PnoposrroN 5.5. For all integers n>0,

(x + t )' log(x * t,: j. (i) *^- h^-*t"- * o;., L;l '-

for lxl{l, where the left side is equal to 0for x: -1.

Setting x: -l in this formula gives the following beautiful formula.

, l f r ) 1 ) : [ - h , f o r n ] - 0

I for il 10,

taking a= | in (8) gives

).1,)(x+1): i l i l ryrr;r -r.rZoLk |

"

156 STEVEN RoMAN

Conorr,cRv 5.6. For all integers n>0,

i r - r r - l 1 l - ( - l ) " * ' .* Z o L e l n

Proof. Setting x: -l in Proposition 5.5, we obtain

i l l l , - 1 )o : - i ( i ) ( h^ -h . -oX - r )ou:',*t Lk |

' *:.o \rc,/

- -hn-t, (;) ( - r)o +J. (;) (-t)o h^- o

:o*( -1)" i ( : ) r - r ) - i . 1' *Z' \k/ i l t t

: ( - 1 ) " i 1 i f T ) ( - l ) o' ,7t i E'\k/

: ( - 1 ) ' i 1 ( - 1 ) ' f ' - l \' ' ' , ? r i '

- ' \ i - t /

_( -1 ) ' i ( 1 ) r_ , r ,n , ? t \ i /

'

_ ( - 1 ) ' � * t .n

Since ! [ :o [X l ( -1 )u : I? :o (?X- l )o :0 , the resu l t fo l lows. I

6. AN Expuclt FoRuurA FoR rI{E HenuoNtc Loclnnuus

We now turn to the matter of finding an explicit expression for theharmonic logarithms. Although these functions are ideal with regard todifferentiation and antidifferentiation, their expression in terms of powersof x and logx is not so simple. (Although it is elegant.)

With the benefit of hindsight, we set

. f l i 'G) : r" 2 ( - I ) i ( t1, cf \( log x) ' - i ,

j : o

where ( t ) i : t ( t - l ) " ' ( / - i+1) , ( / )o :1 , and. c f i ) a te undeterminedconstants. Then we determine the constants c!/) so that the functions

.f!r(*\ satisfy the definition of the harmonic logarithms. After somestraightforward computations, we are led to the following proposition.

-_l

HARMONIC LOGARITHMS

PnoposmloN 6.1. The harmonic logarithms ),f;){x) areformula

fr', :7', and (2)

Ll)(x): a" ( - 1 )' (t), cf)(log x)' - i,

where (t)t : t(t - l) ' ' ' (t - j + 1 ), (t)o : l, and the constants cf) are uniquelydetermined by the initial conditions

(e)t

tL

r57

giuen by the

for i :g.fo, j +O

"5": {;

( 1 ) ' * ' : { ;

and the recurrence relation (for j>0)

(3 ) nclit : cl - ') + lnl cf! y

The numbers c!,), defined for all integers n and all nonnegative integers7 by conditions (11(3), are known as the harmonic numbers. As we will see,these numbers have some rather fascinating properties. Figure 3 showssome values of the harmonic numbers.

Note that condition (l) of Proposition6.l gives us the Oth row of thematrix in Fig.3, and condition (2) gives us the Oth column. Then we canuse condition (3) to fill in the remaining portion of the matrix. For theright portion, we use condition (3) in its given form. (See Fig.4a.) For theleft portion, we use condition (3) in the form fnl cf!r:ncu)-s!l*1). (SeeFig. ab.)

Before discussing the properties of the harmonic numbers, we shouldsettle the matter of showing that the harmonic logarithms form a basis for

j=o --> . . 0 0 0 0 0 0. . . - r - t - l - l - l - l

-s -?8 -? -+ -r 0- f -8 : - r - l o o-l i- t -* o o o

' . ' - * , -h 0 o o o" ' - # o o o o o

n=0YI 1 1 I I I . . .n r 3 l l ?5 l l2 ...

2 6 t z $

0 1 + * * * " 'n 1 E * r t . . '

8

0 1 f i * * * . . .0 1 S * * * " .0 l E * * * . . '

-,-.....J-#

Columns sum to n Columns approach n

FIc. 3. Values of the harmonic numbers cli).

158 STEVEN ROMAN

Lnr.i,], --

(u)

^(j_ I )

/ '

Lnt.irlr

HARMoNTc LocARrrr{Ms 159

(d) (Column -l\

c 9 \ : - 6 , r

(e) (Lower left hand wedge)

cf) :0 for n -n.

Let us look more closely at the harmonic numbers c!l) of nonnegativedegree n ) 0. Note the beautiful pattern emerging in part (a).

PnoposntoN 6.3. For n>0, the harmonic numbers cf;) hatse the followingproperties:

. . 1 1 I( a ) c L " : h n : t * t + 5 + . + :

. l / . l \ r / , l 1 \ r / I l \c ' , i ' : t * t ( t * r ) * l 1 ' t * , * l ) * . . . * ; \ t + 1 + : )

l l - t / l \ tc l i ' : l * t L t * r ( . t * r , / l

1 t - . t / 1 \ r / I l \ t* ; L t* t ( ,1 * ; )* i ( t * i * i , ) l *1 l - . r / 1 \ r / I 1 \* ;L t * t l t * r ) * r l t * r * z ) * "

| / ' * . . 1) -1.+ - l t + t n l JIn general, for j>0, ,1": i . l c\ i- ' t .

(b) (Knuth)

(c) For each n>0, the sequence cf) forms a nondecreasing sequence in j,which is strictly increasing for n>1. Furthermore, we haue, for eachn 2 o ,

lim clt : n.J + @

160

Hence, the limitn: 1, we have

STEVEN ROMAN

Proof. For part (a), since n>0 andT>0, repeated use of condition (3)of Proposition 6.1, for dillerent values of the indices, gives

1,,,!, : icf

- tt + cf! ,

1 , , , , 1: ; r , l - t ) + ; - c l _ , , t + c f ! ,

| . . . . 1 I .: ; r , l - , t + ; : j c f_ - l t + , -c f : ) )

+c( / ! . ,

:| . . . . I I 1: ; "1 -D + ; t " ; : i ' + f1c l - j ) + ' + i c t i - l t + c | fy ) .

But since cyi:0 forT>0, the conclusion follows. Part (b) can be provedusing Proposition 6.1, but we omit the details.

As for part (c), if n:0 or 1, the result follows easily from Proposi-tion6.2. For each n>I, we proceed by induction on j. First, we havecl l t :h,> 1:clo). Assuming that cf - t )>c9- ') , part (a) gives

, y , : i I c v - ' t t i 1 g \ i - . ) - � s , . � r ),?t i ,?t i

and so c!/) is strictly increasing. Furthermore, for a fixed n, c!/) is bounded,as can be seen by using part (b):

l c t ) l ( i f1 ) , - ,< i P \ : r " - t ., ? r \ i ) ' ? ' \ i /

Sn:lim;-- c!/) must exist and be finite. For n:0 and

so:rt1 ,[r):r.LT 6;o: o

and

Sr : l im c ! " ' l : l im I :1 ., t + @ j - q

Let us assume that S,- r:n- 1. Rewriting condition (3) of Proposi-tion 6.1, and taking limits, we have

lim (ncf) - cl - t ') : l i- lnl cf\r : Ln I S, - r : n(n - l).j - q j - @

HARMOMC LOGARITHMS

But since the appropriate limits exist, this can be written

lim ncp - lim cf - 1) : n(n - l)j - e j - q

or

nS, - ,Sn:n(n- l ) ,

from which it follows that ,S,: 12. I

To discuss the properties of the harmonic numbers cf) of negative degreen < 0, we first recall some basic facts about the Stirling numbers s(n,7) ofthe lirst kind. These numbers are defined, for all nonnegative integers n and7, by the condition

x ( x - 1 ) ' . . ( x - n + l ) : i s ( n , j ) x i .j : o

It can be shown that the Stirling numbers of the first kind are characterizedby the following conditions (see [1, p.2l4f):

s(n, 0) : s(0,,1):0, except that s(0, 0): 1

s (n , i ) : s (n - 1 , / - 1 ) + (n - l ) s (n - l , i ) . (10)

Now we can state the following proposition.

Pnoposrtox 6.4. For n0,

f - r 1 I, 1 , : _ l d , , + I ; " j , - " 1 ,L i : n + t t I

where the sum on the right is 0 if n: -1.

161

r62

(b )

(c)

STEVEN ROMAN

tf): (-l)i lnl! s(-n, j), where s(n, j) are the Stirling numbers of thefirst kind.

For eqch n -n, and so only a finite numberof the cf) are nonzero. Furthermore, we haue

L rY ' - | cu) :n .j : o - / : 0

(Contrast this with part (c) of Proposition 6.3.)

Proof. Part (a) can be proved by iteration, in a manner similar tothe proof of part (a) of Proposition 6.3. Part (b) can be proved usingProposition 6.1, with the help of Eqs. (10). As for part (c), the hrststatement follows from Proposition 6.2. For the second statement, we startfrom the expression, valid for n < 0,

x ( x - t ) . . . ( x + n + D : f s ( - n , j ) r ' : ! , / , - t ) i c ] \ x j .i=o Sn l l i7o

Setting x: -I, we get

( - 1 x -2) " ' t ' t : * - i " ! l ' 'L n I . i : oBut by Proposition 3.3, for n < 0,

( - 1 X - 2 ) . . . ( n ) l n l l : ( - l ) ' L - n l ! L n l ! : ( - 1 ) " ( - 1 ) ' L n f : n ,

from which the result follows. I

7. CowcruoNc RrIra,c,ms

We have merely scratched the surface in the study of the algebra L andits differential operators. For example, the harmonic logarithms A',i'U)have a very special relationship with the derivative operator, spelled out inthe definition of these functions. Loeb and Rota show that there are other,at least formal, functions that bear an analogous relationship to otheroperators, such as the forward difference operator / defrned by /p(x):p(x + l) - p(x).The functions associated with the operator / are denotedby (")j;') and called the logarithmic lower factorial functions. In general, thesequences pf(x) associated with various operators can be characterizedin several ways, for example as sequences of logarithmic binomial type,satisfying the identity

p?(x + a ) : j. L;l pi?(a) p?-k@)

HARMOMC LOGARITHMS

We hope that the results of this paper justify speaking of the Romancoellicients as a worthy generalization of the binomial coefficients. (Thisis not to suggest that there may not be other worthy generalizations. )It would be a further confirmation of this fact to discover a nicecombinatorial, or probabilistic, interpretation of the Roman coeffrcients,which, as far as I know, has not yet been accomplished.

RBrnnnNcns

1. L. Corr{rer, "Advanced Combinatorics," Reidel, Dordrecht. 1974.2. K. KNopp, "Theory and Application of Infinite Series,', Dover, New york, 1990.3. D. LoEs rNo G.-C. Ror,r, Formal power series of logarithmic type, Adu. Math.1S (19g9\,

1-1 18.4. S. Rorr{,c,N, "The Umbral Calculus," Academic press, New york, 19g4.5. S. RorraN, The algebra of formal seies, Adu. Math. 3l (1979), 3@*329.6. s. RorurN, The algebra of formal series. II. shelfer sequences, J. Math. Anal. Appl. 74

(1980), 120-143.7. S. Rorr.rrN lno G.-C. Rou, The umbral calculus, Adu. Math.27 (197g), 95-lgg.

Printed by Catherine Press, Ltd., Tempelhof 41, B-8000 Brugge, Belgium

163